Mathematics
Why the Laplacian is Everywhere
Have you ever wondered why the Laplacian is so ubiquitous in physics and engineering? I’ve heard many explanations, many of which refer to symmetry considerations, but there is one argument that has to do with the intuitive meaning of the Laplacian itself.
Consider the heat equation:
The left-hand side tells us how quickly the temperature is changing at a given point 𝑥⃗. What would we expect this to depend on? Well, it seems reasonable to think that it would depend on the temperature of the surrounding area. If the neighborhood around 𝑥⃗ is, on average, hotter than at 𝑥⃗, then we would expect the temperature at 𝑥⃗ to rise. Conversely, if the neighborhood is colder than at 𝑥⃗, we would expect the temperature to fall. And that’s exactly what the Laplacian is telling us!
In the one-dimensional case, the Laplacian is simply:
As you may know from calculus or from this article, the second derivative can be expressed as a limit:
If we rewrite the term under the limit, we see that this is proportional to:
This tells us that the second derivative is (proportional to) the difference between the value of the function at the point of interest, 𝑇(𝑥), and the…
